Computation of structural intensity in plates

COMPUTATION OF STRUCTURAL INTENSITY IN
PLATES

KHUN MIN SWE

THE NATIONAL UNIVERSITY OF SINGAPORE
2003


COMPUTATION OF STRUCTURAL INTENSITY IN
PLATES

KHUN MIN SWE

A THESIS SUBMITTED
FOR THE DEGREE OF MASTER OF ENGINEERING
DEPARTMENT OF MECHANICAL ENGINEERING
THE NATIONAL UNIVERSITY OF SINGAPORE
JUNE 2003


ACKNOWLEDGEMENTS

The author wishes to express his profound gratitude and sincere appreciation to his
supervisor Associate Professor Lee Heow Pueh, who guided the work and contributed
much time, thought and encouragement. His suggestions have been constructive and
his attitude was one of reassurance.
The author commenced his studies under the co-supervision of Associate Professor Dr.
Lim Siak Piang, to whom special thanks are given for his guidance and support
throughout the entire work.
The assistance given by staffs in the Vibration and Dynamic Laboratory during the

study is acknowledged and appreciated. Special thanks are also due to staff from the
CITA, for the valuable advice and help.
The financial assistance provided by the National University of Singapore in the form
of research scholarship is thankfully acknowledged.
Finally, the author wants to express thank to those who directly or indirectly provided
assistance in the form of useful discussion and new ideas.

i


TABLE OF CONTENTS

ACKNOWLEDGEMENTS

i

TABLE OF CONTENTS

ii

SUMMARY

v

LIST OF FIGURES

vii

LIST OF TABLES

xiii

1. INTRODUCTION

1

1.1 Overview

1

1.2 Literature Review

2

1.3 Vibrational Power Flow Calculation

4

1.4 Organization of the Thesis

5

2. THE STRUCTURAL INTENSITY CALCULATION

8

2.1 Computation by the Finite Element method

8

2.2 The Instantaneous and Active Structural Intensity

9

2.3 Formulation of the Structural Intensity in a Plate

10

2.4 Comparisons of the Results

13

2.5 Intensity Calculation at the Centroid

14

3. THE STRUCTURAL INTENSITY OF PLATE WITH MULTIPLE
DAMPERS

17

3.1 Introduction

17

3.2 The Finite Element Model

18

3.3 Results and Discussion

18

3.3.1 Effects of excitation frequency

19

3.3.2 Effects of relative damping

20

3.4 Conclusions

21

ii


4. STRUCTURAL INTENSITY FOR PLATES CONNECTED BY
LOOSENED BOLTS

29

4.1 Introduction

29

4.2 Modeling of Energy Dissipated Loosened Bolts

30

4.3 The Plates Joint Model

33

4.4 Identification of Parameters

34

4.5 Disordered Structural Intensity at Plates

35

4.5.1 Effects pf rotational springs and dampers

35

4.5.2 Shear force effects

36

4.6 Identification of the Bolts

37

4.6.1 Effects of the relative damping at the bolts

37

4.6.2 Effects of the additional damper

38

4.7 Conclusions
5. DISTRIBUTED SPRING-DAMPER SYSTEMS AS LOOSENED

40
53

BOLTS
5.1 Introduction

53

5.2 The Finite Element Model

54

5.3 Identification of Parameters

54

5.3.1 Single Point Connection Systems

55

5.3.2 Distributed Spring-dashpot System over a Finite Area

55

5.4 Results and discussion

56

5.4.1 Single Point Connection

57

5.4.2 Distributed connections

59

5.5 Conclusions

60

iii


6. STRUCTURAL INTENSITY FOR PLATES WITH CUTOUTS

69

6.1 Introduction

69

6.2 Plate Model with Cutouts

70

6.3 Disordered Structural Intensity at Plates

71

6.3.1 SI near Cutouts

71

6.3.2 Convergence Study of the Results

72

6.3.3 Other Investigations

73

6.4 Conclusions
7. FEASIBILITY OF CRACK DETECTION

74
83

7.1 Introduction

83

7.2 Modeling the Cracked Plate

85

7.3 Prediction of the Presence of Flaw

86

7.3.1 The Structural Intensity of Plate with Crack

86

7.3.2 Convergence of Results and significance of crack

87

7.3.3 Crack Orientation Effect

89

7.3.4 Crack Length Effect

89

7.3.5 Input power and intensity value comparison

90

7.4 Conclusions

91

8. CONCLUSIONS

102

REFERENCES

104

iv


SUMMARY

The structural intensity or vibrational power flow is investigated using the Finite
Element Method for several plate structures in this thesis. The structural intensity of
plates with multiple dampers is studied to explore the energy flow phenomenon in the
presence of many dissipative elements. The relative damping coefficients of the
dampers have significant effects on the relative amount of energy dissipation at
corresponding sinks while the frequency affects the energy flow pattern slightly.

The damping capacities of joints play an important part in the analysis of the dynamics
of structures. The intensities for plates connected by loosened bolts are computed. The
bolts are modeled by simple mathematical model consists of springs and dampers. The
loosened joint is modeled in two manners, discrete and distributed spring-dashpot
connections. The results indicate that the rotational springs and dampers have major
effects on the structural intensity of the jointed plates. The presence of loosened bolts
can be identified by the intensity vectors for both joint models. Then, the energy
dissipation and transmission at the joints are calculated and their characteristics of are
discussed.

The structural intensity technique has been proposed to describe the dynamics

characteristics of plates with cutout. The significant energy flow pattern is observed
around the cutouts and it is independent of shapes and positions of cutout on the plate.
Convergence study of the finite element results is also performed with different
numbers of elements. Hence, the presence of cutouts in plates is predicted to locate and
estimate by the intensity vectors.
v


Vibration related damage detection methods appear to be capable alternatives for online monitoring and detecting the structural defects. Thus, the feasibility of flaw
detection and identification by the intensity technique is also investigated. A crack in a
plate can be sensed by the diversion of directions of the structural intensity vectors
around the crack boundary in the structural intensity diagrams. The results also suggest
that the feasibility of detecting the long crack is higher than that of a short one.
However, the crack orientation with respect to the energy flow directions is important
to detect the presence of a crack.

vi


LIST OF FIGURES

Fig. 2.1 Plate element with forces and displacements (a) Moment and force
resultants (b) Displacements

12

Fig. 2.2 Structural intensity field of a simply supported steel plate with a point
excitation force and a damper.

15

Fig. 2.3 Structural intensity vectors of a thin Aluminum plate simply
supported along its short edge with an excitation force and an
attached damping element. (Direct calculation)

16

Fig. 2.4 Structural intensity vectors of a thin Aluminum plate simply
supported along its short edges with an excitation force and an
attached damping element. (Interpolated values)

16

Fig. 3.1 The finite element model of plate showing positions of force and
dashpots

24

Fig. 3.2 Structural intensity field for dashpot with damping coefficient of 100
N-s/m at point-1; Excitation frequency 8.35 Hz.

24

Fig. 3.3 Structural intensity field for dashpot with damping coefficient of 100
N-s/m at point-1; Excitation frequency 17.36 Hz.

25

Fig. 3.4 Structural intensity field for two dampers are attached at point-1 and
point-2; Damping coefficient 100 N-s/m each; Excitation frequency

17.36 Hz.

25

Fig. 3.5 Structural intensity field for two dampers are attached at point-2 and
point-3; Damping coefficient 100 N-s/m each; Excitation frequency
17.36 Hz.

26

Fig. 3.6 Structural intensity field for two dampers are attached at point-1 and
point-3; Damping coefficient 100 N-s/m each; Excitation frequency

vii


point-3; Damping coefficient 100 N-s/m each; Excitation frequency
26
17.36 Hz.
Fig. 3.7 Structural intensity field for a damper at point-2, damping coefficient
1000 Ns/m; Next damper at point-1, damping coefficient 100 Ns/m;
Excitation frequency 17.36 Hz.

27

Fig. 3.8 Structural intensity field for a damper at point-2, damping coefficient
1000 Ns/m; Next damper at point-3, damping coefficient 100 Ns/m;
Excitation frequency 17.36 Hz.

27

Fig. 3.9 Structural intensity field for a damper at point-1, damping coefficient
1000 Ns/m; Next damper at point-3, damping coefficient 100
Ns/m;Excitation frequency 17.36 Hz.

28

Fig. 3.10 Structural intensity field for three dampers with different damping
capacity at the excitation frequency of 17.36 Hz; The first damper at
point-1 with damping coefficient of 200 N-s/m; The second damper
at point-2 with damping coefficient of 400 N-s/m; The third damper
at point-3 with damping coefficient 600 N-s/m.

28

Fig. 4.1 (a) Mode shapes of bolted joint model for two beams

42

Fig. 4.1 (b) Schematic diagram of two square plates joined together by two

42

bolts.
Fig. 4.2 The finite element models of two square plates joined together by two
bolts.

42

Fig. 4.3 Structural intensity field for (a) left hand side plate (b) right hand side

plate, joined together by two loose bolts; the excitation frequency
54.63 Hz; spring-dashpot systems at joints are active in all 6 dof.

43

viii


Fig. 4.4 Structural intensity field of (a) left hand side plate (b) right hand side
plate; plates are connected by loosened bolts; 54.63 Hz; springdashpot system is only active in three translational directions.

44

Fig. 4.5 Structural intensity field for (a) the left (b) the right plate; two plates
are joined together by two loose bolts; the excitation frequency
54.63 Hz; spring-dashpot systems are only active in three rotational
directions.

45

Fig. 4.6 Structural intensity field of (a) the left (b) the right plate caused by
bending moment (Moment component intensity fields) (54.63 Hz)

46

Fig. 4.7 Structural intensity field of (a) the left (b) the right plate calculated
form shear force only (Shear component intensity field) (54.63 Hz)

47

Fig. 4.8 Shear forces distributions at (a) the left (b) the right plate. (54.63 Hz)

48

Fig. 4.9 Intensity vectors of the bolted plates at 20 Hz (a) the left (b) the right
plate.

49

Fig. 4.10 Intensity vectors of the bolted plates at 10 Hz (a) the left (b) the right
plate.

50

Fig. 4.11 Structural intensity diagram of the jointed plates at 10 Hz (a) the left
(b) the right plate. (Damping at the upper bolt is increased to reduce
energy rebound from the right hand side plate)

51

Fig.4.12 Intensity field of plates connected by two bolts (a) the left (b) the
right plate; an additional damper is attached at x = 0.4 m and y = 0.3
m in the right plate; Excitation frequency 10 Hz.
Fig. 5.1 The finite element model of plates overlap over a distance of 0.1 m.

52
64

Fig. 5.2 (a) A spring-dashpot system connecting the two plates at the center
point (b) The distributed spring-dashpot systems connecting two

ix


plates over the finite circular area (solid circles show the positions of
spring-dashpot systems).

64

Fig. 5.3 Structural intensity of the plates with a single point attached springdashpots system (Previous FE model) Excitation frequency 20.03 Hz.

65

Fig. 5.4 Structural intensity of the plates with a single point attached springdashpots system. (a1), (a2), (b2) and (b1) show the enlarged views of
the intensities near the bolts. Excitation frequency 20.03 Hz.

66

Fig. 5.5 Structural intensity fields of plates with distributed springs and
dashpots systems. (a1), (a2), (b1) and (b2) show the enlarged views
near the bolts. Excitation frequency 20.03 Hz.

67

Fig. 5.6 Structural intensity of (a) Distributed system and (b) Single system at
Excitation frequency 10 Hz.

68

Fig. 6.1. The finite element model of a plate with a square cutout near the

center.

77

Fig.6.2 The structural intensity of a plate with a cutout at the frequency of
37.6 Hz (near the natural frequency of the first mode) (Fig.4 (b)).

77

Fig. 6.3 Mesh densities around the cutout (a) coarse (b) normal (c) fine (d)
finest

78

Fig. 6.4 The structural intensity field (a) of a plate with a smaller cutout at the
center (b) around the cutout at 37.6 Hz.

78

Fig. 6.5 SI field of a plate at an excitation frequency of 82.06 Hz.

79

Fig. 6.6 SI field of a plate at an excitation frequency of 107 Hz.

79

Fig. 6.7 Structural intensity (a) of a plate with a circular cutout at the center
(b) around the cutout.

80

Fig. 6.8 The structural intensity field of a plate with a square cutout at the

x


edge, at 37.6 Hz.

80

Fig.6.9 The structural intensity field of a plate with a square cutout at the
edge, at 14.29 Hz. The plate is simply supported along the two
opposite short edges.

81

Fig. 6.10 The structural intensity field of a plate with a square cutout at the
edge, at 28.13 Hz The plate is simply supported along the two
opposite short edges.

81

Fig.6.11 SI field of plate having a cutout with a damper at (0.8 m, 0.15 m), 28
Hz.
Fig. 7.1 The basic finite element model of a cracked plate.

82
93

Fig. 7.2 A higher density FE meshes around the crack and showing the
positions of the source by ‘ ’ and the sink by ‘ ’ for Fig.7.6 (a-e) and
Fig.7.7.

93

Fig. 7.3 Structural intensity field of the whole plate with a vertical crack (51
Hz).

94

Fig. 7.4 Structural intensity around the vertical crack showing the changes in
directions of intensity vectors at the crack edge (51 Hz).

94

Fig. 7.5 Structural intensity around the crack for the model with reduced
numbers of elements. (51 Hz)

95

Fig. 7.6 Comparison of two results from two different FE models at four
particular points (a) Fig.5 and (b) Fig. 7(a).

95

Fig. 7.7 Structural intensity vectors around the crack; the crack is located
between the source and the sink at (a) the first (b) the second (c) the
third (d) the forth (e) the fifth (closest) positions given in Table 1. (51
Hz)

96

xi


Fig. 7.8 Structural intensity around the crack; the source and the sink are
vertically located and parallel to the line of crack. (51 Hz)

98

Fig. 7.9 Structural intensity field of the whole plate with a horizontal crack (51
Hz).

99

Fig. 7.10 An enlarged view of intensity vectors changing their direction at the
crack. (51 Hz)

99

Fig. 7.11 Structural intensity vectors near a long crack. (51 Hz)

100

Fig. 7.12 Structural intensity vectors near a short crack. (51 Hz)

100

Fig. 7.13 Structural intensity vectors turning around a short crack when the

source and the sink are very close to the crack. (51 Hz)

101

xii


LIST OF TABLES
Table 3.1 Percentage of dissipated energy in a plate with multiple dampers at
the frequency near the first resonance (8.35 Hz)

22

Table 3.2 Percentage of dissipated energy in a plate with multiple dampers at
the frequency near the second resonance (17.36 Hz)

22

Table 3.3 Percentage of dissipated energy in a plate with multiple dampers at
three dampers (17.36 Hz)

23

Table 4.1 Variations of natural frequencies of the system with different
springs’ stiffness

42

Table 4.2 The k and c values for the joint with uniform pressure at the bolt in
the two flexible (z & γ) directions

42

Table 4.3 The k and c values for the joint with uniform pressure in other four
(x, y, α and β) directions

42

Table 5.1 Spring stiffness and coefficients of dashpots for the single springdashpot system

62

Table 5.2 Spring stiffness for distributed spring-dashpot systems

62

Table 5.3 Damping coefficients for distributed spring-dashpot systems

63

Table 5.4 Comparison of powers form velocities at 20.03 Hz

63

Table 5.5 Comparison of powers from integration of SI at 20.03 Hz

64

Table 6.1 The data for finite element models

76

Table 6.2 The x and y components of the intensity values at co-ordinate
(x = 0.325 m, y = 0.525 m )

76

Table 6.3 Input powers and the SI comparison for plate with different cutouts

76

Table 7.1 The positions of the source and the sink along y = 0.3 m line.

92

Table 7.2 The x and y components of the intensity values for convergent study

92

Table 7.3 Input powers and the SI comparison for plates with different cracks

92

xiii


CHAPTER 1
INTRODUCTION

1.1 OVERVIEW

Structural intensity is a subject that has gained considerable interest in recent years.
The use of the structural intensity was introduced first as a quantifier for the structural
borne-sound analysis. Later, it has become a new trend in the dynamic analysis of
structures and machines. When a structure is dynamically loaded, the propagation of
vibratory energy through the structure occurs from the result of interaction between the
velocity and the vibration-induced stress and it is termed the vibrational or structure
power flow.

The structural intensity is defined as the instantaneous rate of energy transport per unit
cross-sectional area at any point in a structure. The structural intensity is a vector and
instantaneous intensity is dependent on time. In order to investigate the spatial
distribution of energy flow through the structure, the time-average of the instantaneous
intensity is determined instead of absolute power and it becomes time independent for
steady state response and it describes the relative quantities of the resultant energy flow
at various positions in a structure.

The structural intensity vectors indicate the vibration source and the energy dissipation
points or sinks as well as the magnitudes and directions of energy flows at any position
of a structure. Therefore, the information of vibrational energy propagating in a
1


structure can be visualized by using the structural intensity plots. The structural
intensity technique enables the solving of problems which are associated with
vibrational energy. In noise reduction problems flexural waves are considered since
bending modes in plate are the most critical for sound radiation in an acoustic field. In
order to control these problems, the understanding of dynamic state and the information
of energy flow of a structure is essential.

The structural intensity technique enables the solving of problems which are associated
with vibrational energy by providing the information of dominant power flow paths and
the determination of locations of the sources and the sinks. The required modifications
can be made in order to control the corresponding problems. The changes of
predominant energy flow path may be obtained by alteration of the location of energy
dissipation or the energy sink. The mechanical modifications and the active vibration
control are also options for controlling the power flow. Furthermore, a structure can be
designed to channel and dissipate the energy as necessary.

1.2 LITERATURE REVIEW

The structural intensity was first introduced by Noiseux [1] and later developed by
Pavic [2] and Verheij [3]. These works were mainly related to the experimental
methods. Pavic [2] proposed a method for measuring the power flow due to flexure
waves in beam and plate structures by using multiple transducers and digital processing
technique. Cross spectral density methods was presented by Verheij [3] to measure the
structural power flow in beams and pipes. Pavic [4] proposed a structural surface

2


intensity measurement to analyze a more general vibration type and a structure with
complicated geometry.

Computation of structural intensity using the finite element method was developed by
Hambric [5]. Not only flexural but also torsional and axial power flows were taken into
account in calculating the structural intensity of a cantilever plate with stiffeners. Pavic
and Gavric [6] evaluated the structural intensity fields of a simply supported plate by
using the finite element method. Normal mode summations and swept static solutions
were employed for computation of structural intensity fields and identifying the source

and the sink of the energy flow. The use of this modal superposition method was
further extended as experimental method by Gavric et al. [7]. Measurements were
performed by using a test structure consisted of two plates and the structure intensity
was computed.

Li and Li [8] calculated the surface mobility for a thin plate by using structural
intensity approach. Structural intensity fields of plates with viscous damper and
structural damping were computed using the finite element analysis. The first effort to
use the solid finite elements to compute the structural power flow was performed by
Hambric and Szwerc [9] on a T-beam model. Measurements of the structural intensity
using the optical methods were discussed by Freschi et al. [10] and Pascal et al. [11]. A
z-shape beam was used in order to analyze the propagation of all types of wave in
measuring the structural intensity [10]. Laser Doppler vibrometer was employed to
measure vibration velocities of the beam. Pascal et al. [11] presented the holographic
interferometry method to obtain the phase and magnitude of the velocities of beam and
plates. The structural intensity of a square plate with two excitation forces was

3


calculated in wave number domain and divergence of intensity was computed to
identify the position of the excitation points. Rook and Singh [12] studied the structural
intensity of a bearing joint connecting a plate and a beam.

The active and reactive fields of intensity of in-plane vibration of a rectangular plate
with structural damping were studied by Alfredsson [13]. Linjama and Lahti [14]
applied the structural intensity technique to determine the impedance of a beam for
determination of the transmission loss of general discontinuities.

1.3 VIBRATIONAL POWER FLOW CALCULATIONS

Several analytical methods have been used to predict the energy quantities of vibrating
structures. Modal analysis, finite element analysis, boundary element analysis, spectral
element method, statistical energy analysis, and vibrational power flow method are
mainly used in solving the vibration related problems.

The power flow in two Timoshenko beams was computed by Ahamida and Arruda [15]
using spectral element method for higher frequencies. The statistical energy approach
was employed to investigate the rotational inertia and transverse shear effect on
flexural energy flow of a stiffened plate structure at sufficiently high frequency [16].
The statistical energy analysis (SEA) is mainly employed for the simulation of the
behavior of a structural-acoustic system at high frequencies. SEA uses the total
energies associated with each subsystem of a structure as primary variables. The
vibrational power flow method (VPF) was introduced by Yi el al. [17] and it involves

4


the division of a structure into substructures as in SEA. However, this approach can be
used for both high and low frequencies and the power transmission in beam-plate
structures with different isolation components were computed. BEM is more widely
used in the prediction of the interior noise levels due to structure-borne excitation [18,
19].

The finite element computations in the structural intensity predictions were reported in
references [5-7]. The advantage of calculation of power flow by FEM is that the
available information can be rearranged so that dominant paths of energy transmission
through a structure can be visualized. This procedure seems to have a strong potential
alternative for studying low-frequency structure-borne sound transmission at an early
design process. The finite element method has been used for all the computations of

the structural intensity fields in this study.

1.4 ORGANIZATION OF THE THESIS

Most of the previous works are confined to the determination of structural intensity
over some basic structures such as beams, pipes and simple plates. The evaluations of
energy propagation in the presence of wave reflections in discontinuities such as
several joint types between plates and in changes in thickness or cross-sections are still
needed.

Furthermore, the effects of mechanical modification on the energy flow

distribution are also vital to control the power flow. One further goal of the applications
of the intensity could be the flaw identification in plate structures.

5


This thesis presents the contribution on the mechanical modification of the plate
structure by attaching multiple dampers and the use of the structural intensity technique
for several applications. Three types of applications of structural intensity technique,
the identification of the loosened bolts at the jointed plates, the identification of cutout
in plates and the detection of flaws in plates are investigated in details and the
implications of results obtained in these applications are discussed.

In the first chapter, the principle of the structural intensity is introduced and a literature
review of the structural intensity has been given. Various methods for the power flow
determination are presented briefly. The last part of this chapter offers the organization
of the thesis.

Chapter 2 gives the description of the finite element method to the computation of
structural intensity. The definition of structural intensity and the formulation of
structural intensity for a plate using shell elements are given. The results of the present
study are validated by the two published results available in the literatures.

Chapter 3 presents the effects of multiple dampers on the structural intensity field of an
excited plate.

Chapter 4 gives the structural intensity of plate structures connected by loosened bolts
and subjected to forced excitation. The loosened bolts are modeled by spring and
dashpot systems. Numerical results are presented for the plates connected by two
loosened bolts. The effects of rotational springs and dampers on the structural intensity

6


diagram were discussed. The effects of relative damping at the bolts and the additional
damper at the compliance plate on the energy transmission of joint are presented. The
presence of loosened bolts can be indicated by the intensity vectors at the
corresponding points using the translational springs and dashpots.

Chapter 5 presents the modeling of the loosened bolts connected to the plates by using
the simple distributed spring-dashpot system over the bolts areas.

In chapter 6, the structural intensity of rectangular plates with cutouts is investigated.
The effect of the presence of cutouts on the flow pattern of vibrational energy from the
source to the sink on a rectangular plate is studied. The effects of cutouts with different
shape and size at different positions on structural intensity of a rectangular plate are
presented and discussed.

In chapter 7, the structural intensity of plates with cracks is investigated and Chapter 8
is the conclusions for this thesis.

7


CHAPTER 2
THE STRUCTURAL INTENSITY COMPUTATION

2.1 COMPUTATION BY THE FINITE ELEMENT METHOD

The finite element computation of structural intensity was reported in references [5-7].
Different finite element analysis software packages were applied for calculating the
field variables of the model. The commercial FEM code NASTRAN was employed in
the works [4, 8]. The calculations in reference [8] was carried out by using FEM
software ANSYS. The commercial finite element analysis code ABAQUS [20] has
been used for all the analysis in this study. All the FE models in this study have been
generated by commercial finite element preprocessor program PATRAN 2000. The
structural intensity values are computed and plotted by the Matlab 6.2 environment.

The steady state dynamic analysis procedure has been employed to obtain the
magnitude and phase angle of the response of a harmonically excited system.
ABAQUS provides the responses of structure in the complex forms. The calculation of
steady-state harmonic response is not based on the model superposition but is directly
computed from the mass, damping and stiffness matrices of the model. Though it is
more expensive in terms of computation, it can give more accurate results since it does
not require modal truncations.

The plates are modeled by 8-node thick shell elements with reduced integration points
using all six degrees of freedom per node. It was found that this type of elements is the

8


most appropriate element since transverse shear force effect is taken into account in
this element. This type of elements is designated as S8R in ABAQUS.

2.2 THE INSTANTANEOUS AND ACTIVE STRUCTURAL INTENSITY

Vibrational energy flow per unit cross-sectional area of a dynamically loaded structure
is defined as the structural intensity and it is analogous to acoustic intensity in a fluid
medium. The net energy flow through the structure is the time average of the
instantaneous intensity and the kth direction component of intensity at can be defined as
[6]:
I k =< I k (t ) >=< −σ kl (t )Vl (t ) >,

k , l = 1, 2,3

(2.1)

where σ kl (t) is the stress tensor and Vl (t ) is velocity in the l-direction at time t; the
summation is implied by repeated dummy indices; <…> denotes time averaging.
For a steady state vibration, the complex mechanical intensity in the frequency domain
is given as [7],

1
~
~
C k = − ∑ σ~klVl * = a k + irk
2

(2.2)

Here, the superscript ~ and * denote complex number and complex conjugate and і is
the imaginary unit. Negative sign is used for stress orientations.
The real and imaginary parts of the complex intensity, a k and rk are named the active
and reactive mechanical intensities. The active intensity displays the information of the
energy transported from the source to the parts of the structure where energy is
dissipated. The reactive part has no definite physical meaning, and is regarded as the
reactive intensity and it has no contribution of the net intensity.

9


The active intensity is equal to the time average of the instantaneous intensity and
offers the net energy flow. Therefore, I k is formed as,
~
I k = ℜ(C k )

(2.3)

ℜ(−) stands for the real part of the quantity within the bracket. The intensity
corresponds dimensionally to stress times velocity; thus the unit for structural intensity
is N/ms, the same as that for acoustical intensity.

Power inputs to a system are computed by multiplying input forces by the complex
conjugates of the resulting velocity at the loads points. The total input power due to
point excitation forces can be calculated as
Pin =

1 ⎡ n ~ ~* ⎤
ℜ ⎢∑ F jV j ⎥
2 ⎣ j =1
⎦

(2.4)

where Fj corresponds to load and n is numbers of loads

The power output is the power dissipating through the dampers and transmitting to the
connecting systems such as spring or mass elements. It can be calculated by
Pout =

1 ⎡ n ~ ~* ⎤
ℜ ⎢∑ F j V j ⎥
2 ⎣ j =1
⎦

(2.5)

where Fj corresponds to the force of constraint and n is numbers of attached points

2.3 FORMULATION OF THE STRUCTURAL INTENSITY IN A PLATE

The structural intensity in the plates can be calculated from the stresses and velocities.
Rewriting the equation 2.3 in the form

[

1

~
I k = − ℜ ∑ σ~klVl *
2

]

(2.6)

10